Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves
Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves
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单个椭圆曲线的 Birch 和 Swinnerton-Dyer 猜想的计算验证
作者：
G. Grigorov;Andrei Jorza;Stefan Patrikis;W. Stein;C. Tarnita
我们描述了用于验证解析秩为0和1的Q上特定椭圆曲线的Birch和Swinnerton-Dyer猜想公式的定理和计算方法。我们应用我们的技巧证明了，如果E是一个非CM椭圆曲线在Q上的导体< 1000和秩0或1，那么Birch和Swinnerton-Dyer代数公式的领导系数的L-系列是正确的E，直到奇素数，除了Tamagawa数的E或一些合理的循环issues与域E的程度。由于Birch和Swinnerton-Dyer猜想的秩部分是解析秩为0或1的曲线的定理，这完全验证了这些曲线的全部猜想，直到上面排除的素数。
We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor < 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the L-series is true for E, up to odd primes that divide either Tamagawa numbers of E or the degree of some rational cyclic isogeny with domain E. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely verifies the full conjecture for these curves up to the primes excluded above.